In a small bowl, mash the goat cheese and milk together until smooth. Mix in the garlic, parsley, oregano, and chile flakes. Stir in the olives and season with salt and pepper.
On the thickest side of each breast, cut a deep, 3-inch-long pocket. Using your fingers, stuff the goat cheese mixture into each pocket. Close by pressing the flesh together, securing with a toothpick if necessary.
In a large frying pan, heat the oil over medium-high heat. Have ready a lid that's too small for the pan but will cover all the breasts. Cook the chicken on one side until golden brown, 5 to 6 minutes. Turn the breasts over, season with salt and pepper, and set the small lid on top of them. Continue to sauté until the chicken is cooked through, about another 10 minutes.
Transfer the chicken to a warm serving plate. Pour the wine into the pan and cook, scraping up the flavorful brown bits stuck to the pan, until it's reduced to a glossy syrup. Drizzle the reduction over the chicken and serve.